Since \(\Vert f\circ \varphi _w\Vert _\mathcal {B}=\Vert f\Vert _\mathcal {B}\)  [1]}, and \(f\circ \varphi _w(0)=f(w)\) , it follows that
\(\Vert \mathcal {H}_{K,\mu } f\Vert ^{\prime }_{\mathcal {B}}\le \int _{\mathbb {D}}|K(w)|\Vert f\circ \varphi _w\Vert ^{\prime }_{\mathcal {B}}d\mu (w)=\) 
\(\Vert f\Vert _{\mathcal {B}}\int _{\mathbb {D}}|K(w)|d\mu (w)+\int _{\mathbb {D}}|K(w)||f(w)|d\mu (w).\) 
